Geometry is a branch of
mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning
lengths,
areas, and
volumes, with elements of a formal mathematical science emerging in the West as early as
Thales. By the 3rd century BC geometry was put into an
axiomatic form by
Euclid, whose treatment—
Euclidean geometry—set a standard for many centuries to follow.
Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern
integral calculus. The field of
astronomy, especially mapping the positions of the
stars and
planets on the
celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. Both geometry and astronomy were considered in the classical world to be part of the
Quadrivium, a subset of the seven
liberal arts considered essential for a free citizen to master.